Loops in AdS from conformal field theory

Journal of High Energy Physics, Jul 2017

Abstract We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual 1/N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1/N 2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of ϕ 4 theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving the four-point scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the \( \mathcal{N} \) = 4 super-Yang-Mills theory.

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Loops in AdS from conformal field theory

Received: May Loops in AdS from conformal eld theory Ofer Aharony 0 1 3 6 7 Luis F. Alday 0 1 3 4 7 Agnese Bissi 0 1 3 5 7 Eric Perlmutter 0 1 2 3 7 Andrew Wiles Building 0 1 3 7 Radcli e Observatory Quarter 0 1 3 7 0 Harvard University , Cambridge, MA 02138 U.S.A 1 Woodstock Road , Oxford, OX2 6GG , U.K 2 Department of Physics, Princeton University 3 Rehovot 7610001 , Israel 4 Mathematical Institute, University of Oxford 5 Center for the Fundamental Laws of Nature 6 Department of Particle Physics and Astrophysics, Weizmann Institute of Science 7 Jadwin Hall , Princeton, NJ 08544 U.S.A We propose and demonstrate a new use for conformal eld theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical di culties in direct calculations. We revisit this problem, and the dual 1=N expansion of CFTs, in two independent ways. The rst is to show how to explicitly solve the crossing equations to the rst subleading order in 1=N 2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of 4 theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving the four-point scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an in nite series of poles, and discuss applications to more complicated cases such as the N = 4 super-Yang-Mills theory. AdS-CFT Correspondence; Conformal Field Theory; Gauge-gravity corre- 1 Introduction 1.1 1.2 Setup Summary of results 2 Crossing symmetry in the 1=N expansion 2.1 Setup 2.2 Implications from crossing at order 1=N 4 2.2.1 2.2.2 Truncated solutions at order 1=N 2 Solutions with in nite support at order 1=N 2 3 Loop amplitudes in AdS Mellin amplitudes UV divergences and freg Examples Large spin and the lightcone bootstrap 3.4 Enter crossing 4.1 The Casimir method 4 Solving the one-loop crossing equations Basic idea The basis h(n)(v) Harmonic polylogarithms and the basis h^(n)( ) 0(2;`) due to individual conformal blocks 3.1 3.2 3.3 4.2 4.3 5.1 Tree-level One-loop 3.3.1 3.3.2 3.3.3 4.1.1 4.1.2 4.1.3 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 Summary 4 in AdS 5 Explicit examples Expectations from the bulk Solution from crossing Comparison to AdS results Relation to lightcone bootstrap More general contact interactions 5.2 The four-point triangle diagram of AdS 3 + 4 theory 6 Computing anomalous dimensions from Mellin amplitudes 6.1 6.2 6.3 General problem Application: 0(2;`) in 4 A remark on 3 theory { i { 40 43 44 7.1 7.2 Generalizations Future directions A.1 Degeneracies B Explicit expansions C D 3 OPE data A Operator content of the one-loop crossing equations General expectations for UV divergences and the large n limit of (2) n;` coupled theories of gravity in anti-de Sitter (AdS) space and conformal eld theories (CFTs) with many degrees of freedom (\large N "). Perhaps the most fundamental element in the holographic dictionary is that the AdS path integral with boundary sources is the generating function of dual CFT correlation functions, thus making predictions for large N , typically strongly coupled, dynamics. The 1=N expansion of the CFT correlators maps to the perturbative expansion of AdS amplitudes, which is computed via the loop expansion of Witten diagrams [1{3]. Such basics may seem hardly worth stressing: conceptually, the AdS side of this story appears rather straightforward, and no di erent from at space. However, perhaps surprisingly, the physical content of the AdS perturbative expansion is poorly understood. Beyond tree-level, the computation of AdS amplitudes is nearly unexplored, as almost nothing has been computed. At one-loop and beyond, technical challenges inhibit brute force position space computations: simple one-loop diagrams whose at space counterparts appear in introductory quantum eld theory courses, like the three-point scalar vertex correction and the four-point scalar box diagram, have not been computed in AdS. Even at tree-level, the original calculations [4{13] were impressive but arduous, and struggled to make manifest the relation to CFT data; only recently have leaner, more transparent methods been introduced, including Mellin space [14, 15] and geodesic Witten diagrams [16]. We emphasize that these are not related to challenges of coupling to gravity: in an AdS e ective eld theory sans gravity, that can be dual to a decoupled sector of some CFT [17], the same issues are present. modern methods. The purpose of this paper is to initiate a systematic exploration of loop amplitudes in AdS, and of the dual 1=N expansion of holographic CFT correlation functions, using There are (at least) two main reasons why one might be interested in this problem. The tures: they relate loops to trees [18, 19], gravitational theories to gauge theories [20], and have suggested a re-imagination of the role played by spacetime itself [21]. One is led to ask: what is the organizing principle underlying the structure of AdS scattering amplitudes? Given the existence of a well-de ned at space limit of AdS (Mellin) amplitudes [15, 22], the aforementioned structures should be encoded in, or extend to, the analogous AdS amplitudes. The second is to better understand the large N dynamics of holographic CFTs. The marvelous universality of holographic large N CFTs is typically only studied at leading order, dual to classical calculations in AdS. But the de nition of a holographic CFT must hold at every order in the 1=N expansion. For instance, a large N CFT whose entanglement entropy obeys the Ryu-Takayanagi formula [23], but not the Faulkner-LewkowyczMaldacena correction term [24], cannot be dual to Einstein gravity coupled to matter. It is the analogous correction that we would like to understand about the CFT operator product expansion (OPE) data: namely, what the loop-level constraints are on operator dimensions and OPE coe cients due to the existence of a weakly coupled gravity dual. In addition, for given holographic CFTs whose planar correlation functions are known, we would like to understand how to go to higher orders in the 1=N expansion. While there is some work on one-loop AdS amplitudes [15, 25{27], some of which we will make contact with later, loop physics in AdS has mostly been studied using other simpler observables, speci cally the partition function (e.g. [28{33]). Interesting constraints can indeed be extracted from the one-loop partition function | for example, in a fourdimensional CFT, 1=N corrections to a and c can be computed by adding Kaluza-Klein { 2 { contributions to the Casimir energy in global AdS5 | but correlation functions are much richer objects. In particular, they depend on OPE coe cients and coordinates, and can hence access Lorentzian regimes of CFT. Knowing loop amplitudes in a given bulk theory would open the door to non-planar extensions of dynamical aspects of holography and the conformal bootstrap [34{50]. While AdS loop amplitudes apparently pose di cult technical problems in position space, there is reason for optimism. From the AdS point of view, given a classical e ective action corresponding to the leading order in the 1=N expansion, one extracts the Feynman rules, and computes loop diagrams accordingly. In this sense, loop amplitudes are xed upon knowing all tree amplitudes, in principle. More precisely, the results of loop computations are uniquely determined up to the need to x renormalization conditions for some parameters; for any theory, renormalizable or not, only a nite number of conditions is required at any given loop order. The problem is to make the relation between loop-level and tree-level AdS amplitudes precise, a la the Feynman tree theorem and generalized unitarity methods for S-matrices. How can quantitative progress be made? We will show that analytic solutions of the conformal bootstrap for these four-point functions may be found at subleading orders in the 1=N expansion. This may be viewed as either a CFT or a bulk calculation. The leading order solutions for the connected four-point function of a single Z2-invariant scalar primary were constructed in [51], where they showed that there is a one-to-one mapping between those solutions and classical scalar eld theories on AdS space with local quartic interactions. An important technical simpli cation of the leading order solutions is that they have nite support in the spin, which makes manifest the analytic properties of the four-point correlator. At subleading order this is no longer the case and the method of [51] does not apply. Nevertheless, solutions can be constructed as a systematic expansion around large spin, adapting the machinery of [52{54]. We nd that the solution to order 1=N 4 is fully xed in terms of the data to order 1=N 2, to all orders in the inverse spin expansion. Likewise we will show that the Mellin representation of the CFT four-point functions makes it clear why and how higher orders in 1=N are determined by the leading-order result. Furthermore, we will reconstruct the full one-loop Mellin amplitude for several examples. 1.1 Setup Throughout the paper, we study an identical-scalar four-point function, hOOOOi, for a scalar primary O of dimension . This is determined in terms of an \amplitude" G(u; v) of the two conformal cross-ratios, u and v (more details will be given in the next section). G(u; v) admits an expansion in 1=N : (0) is determined by mean eld theory, while G (1) G (2) G1 loop are the bulk tree-level and one-loop amplitudes, respectively.1 At every Gtree and 1Following [51], we use the large N gauge theory notation 1=N 2 to stand for the small coupling in the bulk. In a generic, full- edged holographic CFT, this stands for 1=c (even when c does not scale as N 2). { 3 { (1.2) (1.3) order in the 1=N expansion, the amplitude is subject to the crossing equation v G (i)(u; v) = u G (i)(v; u) : We will also work with the Mellin representation of this amplitude, M (s; t), which admits an analogous expansion. Any large N CFT containing O also necessarily contains a tower of \multi-trace" primary operators that are composites of O. The most familiar of these are the doubletrace operators [OO]n;`, one for each pair (n; `) [15], whose de nition we recall below. These acquire corrections to their conformal dimensions n;` and squared OPE coe cients OO[OO]n;` , at every order in the 1=N expansion: to the one-loop Mellin amplitude M1[OOlo]op. Mean eld theory determines a(n0;`), and the tree-level crossing equation determines n(1;`) and a(n1;`) [51]. To solve the one-loop crossing equation is to derive n(2;`) and a(n2;`), in addition to the OPE data for any other operators appearing at that order. We call the [OO] contributions 1.2 Summary of results dimensions n(1;`) In [51], the authors considered generalized free eld sectors of holographic CFTs in which the only operators appearing at O(1=N 2) are the [OO] double-trace operators. Such setups are dual to the simplest e ective eld theories in AdS, namely, 4-type theories with no cubic couplings. We will sometimes call these theories \truncated" theories on account of the spin truncation 2p derivatives at the vertices has L = 2b p2 c. (1) n;`>L = 0 for some nite L, as used in [51]; a bulk theory with We note that in a truncated theory, the double-traces are the only contributions to the full M1 loop: even at O(1=N 4), the O OPE contains no single-trace operators by design, and no higher multi-trace operators by O necessity (a fact which we explain in appendix A). One advantage of Mellin space is that it allows us to show explicitly how M1[OOlo]op can in principle be derived directly from 1=N considerations alone. We show how large N xes the poles and residues of M1[OOlo]op, for any theory, in terms of the tree-level anomalous . The location of the poles has been understood, and the residues derived in a speci c example, in [15, 26, 27]; we show how to obtain this in general. We derive the leading residue explicitly for a general theory (see (3.26)). In a truncated theory, the leading residue is su cient to determine the large spin asymptotics of passes a check against the lightcone bootstrap [35, 36] as applied to 4 theory. (2) n;` . The latter In a general bulk theory such as 4, this stands for the four-point couplings (such as ), while three-point bulk couplings scale as 1=N . We will freely interchange the labels tree-level/ rst order/O(1=N 2) throughout the paper, and likewise for one-loop/second order/O(1=N 4). { 4 { However, the above approach is somewhat clunky to implement and is not maximally physically transparent. A more elegant, and more practical, approach is to solve the oneloop crossing equations for n(2;`) and an;`. This is tantamount to knowing the dual AdS (2) one-loop amplitude. The statement of bulk reconstruction is not just philosophical: we can actually reconstruct M1 loop from OPE data, because they are related by a linear Mellin integral transform. This is our proposed use for crossing symmetry: given leading order OPE data, we solve the crossing equations at the next order, thus reconstructing M1 loop for the dual AdS theory. Let us now discuss what is involved in actually solving the loop-level crossing equations. At one-loop, the tree-level data acts as a source in the crossing equation for G(2)(u; v), which has a unique inhomogeneous solution. The freedom to add a homogeneous solution matches expectations from the bulk, where one is free to modify the local quartic couplings at every loop order: from [51], the correspondence between local quartic vertices and homogeneous solutions to crossing follows. This pattern continues at higher orders. In this work, we will focus on the anomalous dimensions (2) n;` . To actually compute these from crossing, our main observation may be sketched as follows. In the regime u v 1, G(2)(u; v) contains terms of the form xed by lower-order data, and is quadratic in the rst-order anomalous dimensions n(1;`) (hence the log2(u)). By crossing symmetry, we also have contribution to G the precise equation is At this stage we specify to truncated theories, where fan;`; n;` g vanish above some nite `. It is easy to see from the small u expansion that the term in (1.5) must come from a (1) (1) (2)(u; v) that is linear in (2) n;` . For n = 0, where the analysis is simplest, (1.4) (1.5) (1.6) X a(00;`) 0(2;`)g2coll +`;`(v) = 2f (v) log2(v) + ` where gcoll 2 +`;`(v) is the lightcone, or collinear, conformal block, and \ " denotes logarithmically divergent or regular terms. This is the desired equation for 0(2;`) in terms of rst-order data 0;` . (1) nds 0;` (2) The solution of (1.6) is performed order-by-order in the large spin expansion: because each term on the left-hand side diverges like log v, it must be that and its large spin behavior is determined by matching to f (v). At leading order, one 0(2;`) 6= 0 for all `, ` 2 . A systematic expansion requires further development of the Casimir methods utilized in [52{54], adapted now to this particular one-loop equation. Given a large spin expansion of 0(2;`), a resummation down to nite spin is possible when Altogether, both the large and nite spin data constitute a holographic construction of the one-loop amplitudes in the dual AdS theory that classically gives rise to the 0(1;`) used in the crossing problem. { 5 { In the above large spin analysis, we encounter an exciting mathematical surprise: a certain class of harmonic polylogarithms forms a basis of solutions. In particular, if we expand (` + )(` + weight w 0(2;`) to nth order in inverse powers of the collinear Casimir eigenvalue J 2 = 1), then for integer > 1, f (v) can be written as a linear combination of 2 + n harmonic polylogs, de ned in (4.16){(4.19).2 Harmonic polylogs are speci ed by a weight vector, and only a speci c subclass of such functions appears in our problem, namely those speci ed by the alternating w-vector ~w = (: : : ; 0; 1; 0; 1). Given that multiple polylogs are ubiquitous in one-loop amplitudes in at space, it is intriguing to see some of them appearing in the construction of one-loop amplitudes in AdS via the crossing equations. Before showing our results for speci c theories, we should address an obvious question: what happens when an AdS theory has a UV divergence? In particular, how is this visible in the solutions to crossing? This has a satisfying answer. We expect to be able to cancel UV divergences by adding a nite number of local counterterms to our AdS e ective action at a given loop order, just as in at space. As explained in [51], local quartic vertices with 2p derivatives generate anomalous dimensions only for double-trace operators of spin ` 2b p2 c. Therefore, on account of bulk locality of the divergences, we have a precise prediction: when we compute a divergent one-loop bulk diagram via crossing, n(2;`) should diverge for the above range of spins, where p is the number of derivatives in the counterterm. Moreover, for any regularization, the divergence should be proportional to n(1;`). Analogous statements apply at any loop order. We demonstrate all of the above explicitly in the following two examples: 1) 4 in AdS. The only non-trivial one-loop diagram is the bubble diagram of gure 1. This is the one case where M1 loop is actually known directly from a bulk calculation, performed in Mellin space in [26]: the authors used an AdS analog of the Kallen-Lehmann representation to write the loop as an in nite sum of trees. Using our large spin data, we reconstruct this amplitude in AdS3 and AdS5 for a = 2 scalar, exactly matching the result of [26]. (See (5.10){(5.13).) Moreover, we show how to analytically compute 0(2;`) at some nite spins directly from M1 loop itself. The results match the resummation of the large spin solutions to crossing. This extraction had not been done previously | indeed, we know of no case in the literature where OPE data has been analytically derived from a Mellin amplitude with an in nite series of poles. We expect our regularization techniques to be useful more widely in the world of Mellin amplitudes. The results at low spin align precisely with our UV divergence expectations. The AdS3 theory is nite, but the AdS5 theory requires a Accordingly, 0(2;0) diverges in d = 4 (AdS5) but not in d = 2 (AdS3), and 0(2;`) for ` = 2; 4 is 4 counterterm. nite in both cases. (See (5.3){(5.8).) 2) Triangle diagram of 3! 3 3 + 4! 4 4 in AdS. This diagram, shown in gure 1, has never been computed, in any bulk spacetime dimension, as the trick of [26] does not 2The basis for 2= Z can be thought of as comprised of analytic continuations of harmonic polylogs to non-integer weight. It would be interesting to formalize this. { 6 { HJEP07(21)36 work here. Taking = 2 for concreteness, we compute large and nite spin anomalous dimensions from crossing (see (5.32){(5.35)), and reconstruct M1 loop. In d = 4, and in the t-channel, say, 4 scalar, and is the rst computation of any such triangle diagram in any dimension. We also give the analogous result in AdS3 in (5.41). It would be very interesting to discover new tools for a direct evaluation in the bulk. Overall, our work takes a step toward the nite N , Planckian regime by illuminating the structure of the perturbative amplitude expansion in AdS and in large N CFT. As we discuss in section 7, we believe that there is potential for the large N bootstrap to address interesting questions beyond the realm of holography. The paper is organized as follows. In section 2, we set up the crossing problem and identify the key one-loop constraint. In section 3, we review the basics of Mellin amplitudes, and use large N alone to explain how M1 loop is constrained by tree-level data, and to construct the leading residue explicitly. In section 4, we develop the necessary tools for solving the one-loop crossing equations in general. In section 5, we apply our machinery to compute the bubble diagram of 4 in AdS, and the triangle diagram of 3 + 4 in AdS, 4 via crossing. In section 6, we explain quite generally how to compute low-spin anomalous dimensions from Mellin amplitudes with an in nite series of poles; as an example, we apply this to the one-loop bubble diagram in . We conclude in section 7 with a discussion of generalizations, applications to full- edged CFTs like N = 4 super-Yang-Mills and the d = 6 (2,0) theory, and other future directions. Some appendices include further details. 2 2.1 Setup Crossing symmetry in the 1=N expansion Consider a generic CFT with a large N expansion and a large mass gap. More precisely, we assume there exists a \single-trace" scalar operator O of dimension , and that all other single trace operators acquire a very large dimension as N becomes large. This is equivalent to considering a weakly coupled theory of a single scalar eld in AdS, with three-point couplings proportional to 1=N and four-point couplings proportional to 1=N 2. Consider the four-point function of identical operators O. Conformal symmetry implies hO(x1)O(x2)O(x3)O(x4)i = G(u; v) 2 ; xj and we have introduced the cross-ratios u Crossing symmetry implies x213x224 x212x234 and v (2.1) xx212143xx222234 . (2.2) v G(u; v) = u G(v; u): { 7 { We would like to study solutions to the crossing equation in a large N expansion, up to O(1=N 4). As discussed in appendix A, up to this order and to inverse powers of the mass gap, the operators appearing in the OPE of O with itself in a generic CFT include O O 1 + O + T + [OO]n;` + [T T ]n;` + [OT ]n;` ; (2.3) where 1 denotes the identity operator, T the stress tensor, and the double-trace operators [OO]n;` are conformal primaries of the schematic form [OO]n;` = O The presence of O and [OT ]n;` is forbidden in a theory with Z2 symmetry. Furthermore, in the simplest setting we can ignore the presence of the operators including the stress tensor; this is a good approximation when the self-couplings of the scalar are much larger than its gravitational couplings, which is true in particular for a non-gravitational theory on AdS. On the other hand, the presence of double-trace operators [OO]n;` is necessary for consistency with crossing symmetry. Note that higher-trace operators will appear at . higher orders in the 1=N expansion, but not at O(1=N 4). Let us for the moment focus on the simplest setting, in which the operators in the OPE include only the identity operator and double trace operators [OO]n;`. This is relevant for computing correlators of a 4 theory in AdS. In this case the four-point function admits the following conformal partial waves decomposition: G(u; v) = 1 + X 1 1 X n=0 ` even an;`u n;` ` 2 g n;`;`(u; v) ; in which only even values of ` appear,3 and an;` denote the OPE coe cients squared of [OO]n;` in the O O OPE. The normalization of O has been chosen such that the contribution of the identity operator is exactly 1. The conformal block for exchange of a dimension p, spin-` primary is written as G p;`(u; v) = u 2 g p;`(u; v) p ` so as to make manifest the leading behaviour for small u. Although most of the methods of this paper will be general, we will mostly focus on d = 2 and d = 4 for de niteness. For these cases the conformal blocks are given by (2.5) with g p;`(z; z) = g p;`(z; z) = z`F p+` (z)F p ` (z) + z`F p ` (z)F p+` (z) 2 2 2 1 + `;0 z`+1F p+` (z)F p ` 2 (z) z`+1F p ` 2 (z)F p+` (z) 2 2 2 2 ; where we have introduced the parametrization u = zz; v = (1 z)(1 z) for the cross-ratios, and F (z) 2F1( ; ; 2 ; z). 3We assume that the identical external operators are uncharged under any global symmetries. Henceforth we leave the even spin restriction implicit, and use P 1 P At zeroth order in a 1=N expansion the four-point correlator (2.1) is simply the sum over the disconnected contribution in all three channels: This is consistent with the expected spectrum for double-trace operators at zeroth order G (0)(u; v) = 1 + u + u v : (n0;`) = 2 + 2n + ` ; and leads to the following OPE coe cients [51] a(n0;`) = 2Cn Cn+`; a(n0;`) = 2(` + 1)(2 + 2n + ` 2) ( 1)2 Cn 1Cn+`1+1; where we have introduced Cn = 2 ( + n) (2 + n 1) (n + 1) 2( ) (2 + 2n 1) : We will study corrections to the four point function in an expansion in powers of 1=N The dimensions and OPE coe cients of double-trace operators will have a similar expansion n;` = (n0;`) + Let us start by recalling the analysis at O(1=N 2). Plugging the expansions for the dimensions and OPE coe cients into the conformal partial wave (CPW) decomposition (2.4) we obtain G (1)(u; v) = (2.16) Due to the convergence properties of the OPE, the right-hand side displays explicitly the behaviour around u = 0. On the other hand, to understand the behaviour around v = 0 is more subtle. Each conformal block behaves as g p;`(u; v) v!0 a~ p;`(u; v) + ~b p;`(u; v) log v ; (2.17) where a~ p;`(u; v) and ~b p;`(u; v) admit a series expansion around u; v = 0. Hence each conformal block diverges logarithmically as v ! 0. However, in nite sums over the spin may generically change this behaviour. This will be important for us below. In [51] a basis of solutions f n;` (1) ; a(n1;`)g to the crossing equation (2.2) was constructed. Each of these solutions has support only for a bounded range of the spin `. In this case, (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) HJEP07(21)36 the analytic structure around both u = 0 and v = 0 is manifest, and the crossing equation v G (1)(u; v) = u G (1)(v; u) can be split into di erent pieces, proportional to log u log v, log u, log v and 1 (times integer powers of u and v). In [51] it was argued that there is a one-to-one map between this basis of solutions to crossing and local four-point vertices in a bulk theory in AdSd+1. Furthermore, let us mention that in Mellin space these solutions correspond simply to polynomials with appropriate symmetry properties. The degree of the polynomial determines for which range of spins the corrections f n;` from zero. Let us stress that these \truncated" solutions are consistent with crossing only (1) ; a(n1;`)g are di erent in the minimal set-up, in which only the identity and double-trace operators are present in the OPE O O. Later we will discuss what happens in more general cases. The aim of the present paper is to extend those solutions to consistent solutions to (1) crossing at order 1=N 4. We will assume the leading order solutions f n;` (1) ; an;`g as given, (2) ; a(n2;`)g. Plugging the expansions (2.14) into the and analyze consistency conditions on f n;` CPW decomposition (2.4) we obtain 8 n;` 12 a(n1;`) n(1;`) log u + ( n(1;`))2 log2(u) + 2 log u g2 +2n+`;`(u; v) : (2.18) (2.19) ; a(n1;`)g. The contribution from the other lines is uniquely solution at order 1=N 2 and can be viewed as a source, or an inhomogeneous term, for the xed in terms of the crossing equation v G (2)(u; v) = u G (2)(v; u) ; interpreted as an equation for f n;` (2) ; a(n2;`)g. The analysis of this equation is much harder than the analysis at order 1=N 2, since, as we will see momentarily, consistency with crossing implies that f n;` (2) ; a(n2;`)g are di erent from zero for arbitrarily large spin. We will focus here on certain unambiguous contributions to the source terms, and understand their implications for the solution to the crossing equation. 2.2 Implications from crossing at order 1=N 4 Let us focus on a speci c contribution to G(2)(u; v), which is the coe cient of log2(u): G This contribution is unambiguously xed in terms of the leading order solution. We can already make the following simple observation. Under crossing symmetry this term will map to a term with a divergence log2(v) as v ! 0. Since each conformal block diverges at most logarithmically in this limit, such a contribution must come from an in nite sum over the spin, for a given twist. Hence it follows that the solution f n;` ; an;`g must be di erent from zero for arbitrarily large spins, even if the solution at order 1=N 2 is truncated. From now on, it is convenient to restrict our considerations to truncated solutions at order 1=N 2. (2) (2) More general solutions will be studied in section 2.2.2. If the solution at order 1=N 2 truncates at spin L, we have: G where we have used the fact that the sum over spins truncates. f (u; v) and g(u; v) admit a series expansion in u; v with integer powers, and can be computed in terms of the given leading order solution. As a consequence of crossing symmetry, G contain the following terms: (2)(u; v) should also G (2)(u; v) = u log2(v) (f (v; u) log u + g(v; u)) + (2.22) where the dots denote contributions proportional to log v, or analytic at v = 0. Given that the support of f n;` (1) ; a(n1;`)g involves a cannot generate a log2(v) behaviour, since each conformal block diverges at most logarithnite range of the spin, the last two lines of (2.18) mically. Hence (2.21) n 12 a(n0;`) n(2;`)g2 +2n+`;`(u; v) log2(v) = f (v; u) ; (2.23) and there is a similar equation involving the OPE coe cients a(n2;`). (2.23) should be interpreted as an equation for n(2;`), with the right-hand side f (v; u) completely xed in terms of the solution at order 1=N 2. As already mentioned, since we need to reproduce an enhanced divergence on the left-hand side, we need to sum over an in nite number of spins. Furthermore, the divergence will arise from the region of large spin. In section 4 we will adapt the algebraic method developed in [53, 54] to determine the necessary large spin behaviour on n(2;`) in order for (2.23) to be satis ed. The nal answer is an expansion of the form n(2;`) = c(n0) `2 1 + b(n1) ` + b(n2) `2 + ! ; (2.24) where all the coe cients of the expansion are actually computable. Hence we conclude that (2.23) actually xes n(2;`) up to solutions which decay faster than any power of the spin. Notice in particular that, from this point of view, we cannot expect to do any better, since there is always the freedom to add to any solution of (2.19) a truncated solution which solves the homogeneous crossing equation (the same equation appearing at order 1=N 2). In section 5 we will study several examples. For these examples we will actually be able to do much more: we will be able to re-sum the whole series (2.24), and extrapolate the results to nite spin. log u + aT v u d 2 log u + 2 ; which lead [35, 36, 55] to the following large spin behaviour for the anomalous dimensions of double-trace operators: (1) n;` (1 + ) + `adT2 (1 + ) : This implies, in particular, that the leading order solution has in nite support in the spin. Single-trace operators can be seen as sources for the crossing equations, which are otherwise homogeneous. The general structure of the solution at order 1=N 2 is then the sum of a solution to the equation with sources, with the behaviour (2.27), plus any of the truncated solutions studied above. Although any full- edged conformal eld theory contains the stress tensor, we will discuss its inclusion in a separate publication. In this paper, instead, we will consider only the presence of O. This is relevant for correlators of 3 theory on AdS. In this case where n(1;`); 3 n(1;`) = a (1);trunc is any one of the truncated solutions. As before, we can compute the piece proportional to log2(u) at order 1=N 4. We obtain G Let us now discuss the more general situation, in which the OPE of O with itself also includes single-trace operators. The two most important examples are O itself, of dimension , and the stress tensor T , a spin two operator of dimension T = d and twist T ` = d 2. These single-trace operators enter in the OPE decomposition with OPE coe cients squared of order 1=N 2. In these cases G (1)(u; v) contains the following terms G (1)(u; v) = a u 2 g ;0(u; v) + aT u d 2 2 gd;2(u; v) + Under crossing symmetry these map into terms of the form where now the sum over ` is not truncated. As we have already discussed, for a truncated solution the small v behaviour is simply proportional to log v, as for a single conformal block. In the case at hand, however, since the sum over the spin now does not truncate, we get an enhanced behaviour. More precisely, (2.27) leads to G Under crossing symmetry this contribution maps to itself, so that h(u; v) = h(v; u). This is a consequence of crossing and the OPE expansion, and is completely independent of the new data f n;` (2) (2) ; an;`g at order 1=N 4. In addition, as in (2.21), the sum above will contain contributions proportional to log v However their computation is more subtle than before: one needs to perform the sum over the spin, and then expand for small v. Both the truncated and non-truncated parts of the solution will contribute to this term. The analysis of the crossing equations is now more complicated. Under crossing the term (2.31) maps to a term proportional to log u log2(v). However, as the support of the solution at order 1=N 2 is in nite, several terms in (2.18) can produce an enhancement log2(v), and not only those involving n(2;`). While this general case can also be analysed, note that the contributions from crossed terms to ( n(1;`))2, of the form (2a (1);trunc) are much simpler to analyse. These crossed terms have a nite support, and their contribution to n(2;`) can be computed exactly as explained above. We will discuss the interpretation of these contributions, and will compute them for speci c examples, in section 5. 3 Loop amplitudes in AdS The subleading solutions discussed in the previous section may be interpreted as one-loop contributions to correlation functions in AdS. We now turn to constraining the general form of loop-level AdS amplitudes by studying features of the large N expansion. We will employ the Mellin representation. One of the advantages of Mellin space is that AdS amplitudes have a transparent analytic structure as a function of the Mellin variables. This has been utilized in [15] to write down compact and intuitive forms for tree-level Witten diagrams, and we will do the same here at one-loop. See [14, 15, 26, 27, 56{62] for foundational work, and [22, 62{68] for some recent applications, of Mellin space in CFT. 3.1 Mellin amplitudes spondence. We now give a crash course in Mellin amplitudes in the context of the AdS/CFT corre Consider the four-point function of identical operators hO(x1)O(x2)O(x3)O(x4)i, related to an amplitude G(u; v) by (2.1). By a double Mellin transform, we can trade G(u; v) for the Mellin amplitude, M (s; t), de ned to be t. The two integration contours run parallel to the imaginary axis, such that all poles of the gamma functions are on one side or the other of the contour.4 The product of gamma functions is totally symmetric in permutations of (s; t; u^). Crossing symmetry of G(u; v) then implies total permutation symmetry of M (s; t; u^): M (s; t) = M (s; u^) = M (t; s) : (3.2) 4The following formulae are specialized to the case of identical external operators, although many also hold for pairwise identical operators. A summary of the relevant formulae can be found in appendix A of [59]. Their conventions are the same as in (3.1) up to a shift shere = sthere + 2 . gcopll;`(v) is the collinear block, The Mellin representation for all g(mp);`(v) is g(mp);`(v) = Second, Q`;0(s; p) takes the explicit form (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) For pairwise identical external operators as here, the Q`;n(s; p) do not depend on the external dimensions. We will make use of the following facts about the Q`;n(s; p). First, they are intimately related to the Mellin transform of the conformal blocks for exchange of a twist- p operator. In the lightcone expansion u 1, the blocks take the form (2.5) with g p;`(u; v) = 1 m=0 X umg(mp);`(v) : gcopll;`(v) = (1 v)` 2F1 p + ` 2 ; p + ` 2 ; p + `; 1 v : In a CFT with a weakly coupled AdS dual, the conformal block decomposition of G(u; v) translates into a sum of poles in M (s; t). In a given channel, say the t-channel, the amplitude M (s; t) has poles in t at the twists of exchanged operators, and the residues encode the OPE coe cients: where the exchanged primary operator Op has twist p = p `p. The pole at t = p + 2n captures contributions of the twist-( p + 2n) descendants of Op: schematically, these are The residues Q`;n(s; p) are the Mack polynomials, whose precise de nition can be found in appendix A of [59]. They have a spin index ` and a \level" n, and they depend on both the external and internal operator data. We will nd it convenient to work with a \reduced" polynomial, Q`;n(s; ), related to Q`;n(s; ) in general by [59] Q`;n(s; p) = Q`;n(s; p) 2 ( p + `)( p 1)` v (s+ p 2 )=2Q`;m(s p; p) 2 2 s 2 2 s 2 : Q`;0(s; p) = 2 ` p 2 2 ` ( p + ` 1)` 3F2 `; p + ` 1; s 2 ; p 2 2 ; p ; 1 : This has the useful property that Q`;0(s p; p) = ( 1)`Q`;0( s; p) ; ` 2 Z : These obey an orthogonality relation [59], which can be written5 Given some amplitude expanded in the lightcone regime of small u and xed v, this relation allows one to strip o the coe cient of the leading-twist, spin-` lightcone block g2coll +`;`(v) due to the exchange of [OO]0;`. We now develop the AdS loop expansion of the connected piece of G(u; v) and M (s; t), corresponding to the 1=N expansion of some holographic CFT: where An(v) = of our knowledge. 1 4 in!2 Z i1 i1 1 1 1 1 To set the stage for M1 loop, we need to review the structure of Mtree. We are interested in paradigmatic large N holographic CFTs which have a large gap in their spectra, or generalized free eld sectors thereof. These are dual to weakly coupled gravity, coupled to a nite number of light elds. The spectra of these theories consist of \single-trace" operators Oi and their \multi-trace" composites [OiOj ]; [OiOj Ok], etc., that are dual to single-particle and multi-particle states in the bulk, respectively. As discussed above, the CFT conformal block decomposition of Gtree only includes single-trace and double-trace exchanges. There are two salient points about Mtree. The rst is that its only poles come from the single-trace exchanges of Gtree. These each contribute as in (3.3). The second is that the double-trace exchanges of Gtree are accounted for by the explicit 2 factors in the Mellin integrand (3.1), one for each channel, which have double poles at = 2 + 2n. This makes explicit a fact about holographic CFTs: at tree-level, the single-trace OPE data completely determine the double-trace OPE data, up to the presence of regular terms in Mtree. The gamma function residues include a log u term and a term regular at small u, (3.10) `;`0 : (3.11) (3.12) (3.13) (3.14) : (3.15) t=2 +2n Gtree(u; v) = u +n An(v) log u + Bn(v) ; Res ds v (s+2n) 2 Mtree(s; 2 + 2n) 2 s + 2n 2 2 2 2 s 5Analogous orthogonality relations exist at higher n but have not been calculated explicitly, to the best The Mellin representation of g2coll +`;`(v) may be written where we have introduced a convenient combination for future use, gcoll 2 +`;`(v) = ( 1)`16 d ;` Upon using the orthogonality relation (3.11), one nds the explicit formula [62] 0(1;`) = A similar analysis allows one to extract a0;` . For higher n, one must deconvolve the subleading corrections g(m;`)(v) to the small u blocks, from the leading contributions coming from n > 0 double-trace primaries. Expressions and an algorithm for computing g(m;`)(v) can be found in [53]. 3.3 fall into two categories: We now turn to M1 loop. In a general CFT, this may receive various contributions. These Bn may be extracted similarly. Matching this to (2.16), one can extract n(1;`) and a(n1;`) by picking o the contribution proportional to the appropriate conformal block in the u expansion. The An log u terms contain n(1;`), and the Bn terms contain a(n1;`). The extraction of the leading-twist double-trace operator data, like 0(1;`), is especially simple: from (2.18), 1 we require First, there are loop corrections to tree-level data. This includes mass, vertex and wave function renormalization of elds already appearing at tree-level; that is, O(1=N 4) changes to the norms, dimensions and OPE coe cients of CFT operators appearing in the planar correlator. Corrections to the OPE data of single-trace operators can arise, but they can be easily taken into account by expanding the leading order solutions, and we will assume for simplicity that they vanish. Note that in any case these cannot be determined by the crossing equations, which have solutions for any such data. Second, as discussed in appendix A, there are new operator exchanges that do not appear at tree-level, due to large N factorization. A simple example in a theory of gravity coupled to a scalar eld is the appearance of two-graviton intermediate states, dual to [T T ]-type double-trace operators, in the scalar correlator hOOOOi. A universal contribution in any holographic CFT is the next-order correction to the tree-level [OO]n;` OPE data, namely, n(2;`) and a(n2;`). Let us write the double-trace piece of the total one-loop amplitude as M1[OOlo]op(s; t) : We note that in simple AdS e ective theories like 4 dressed with any number of deriva tives, this is the full amplitude. More precisely, for any theory in which no single-trace operators appear in the OPE (dual to theories in AdS with no cubic vertices), and in which there are no extra double-trace operators in the OPE (dual to the absence of four-point couplings to other elds in AdS), we have M1 loop(s; t) = M1[OOlo]op(s; t): (3.21) in section 7. (1) n;` and an;`. When f ig 2 2Z in more general theories, there are similar simpli cations, as we discuss We now establish the following simple but powerful claim: all poles and residues of M1[OOlo]op are completely xed by tree-level data. It follows that n(2;`) and a(n2;`) are xed by Recall that the contribution of [OO]n;` to G(2) takes the form given in (2.18): G The point is that there is a log2(u) term whose coe cient is completely xed by tree-level data. In order to correctly produce this term at each power u +n (n = 0; 1; 2; : : :), two things must happen: 1) M1 loop must acquire simple poles at = 2 + 2n for n = 0; 1; 2; : : :. 2) The residues are xed by n(1;`) so as to match (3.22). This is true in each of the s; t; u^ channels, so we can focus on just one, and trivially add the crossed channels to get the full M1[OOlo]op. Showing the t-channel for concreteness, we have thus determined that M1[OOlo]op(s; t) = 1 X n=0 t Rn(s) (2 + 2n) + freg(s; t) + (crossed) (3.23) for some residues Rn(s). This argument does not determine any possible regular terms in M1[OOlo]op, so we have allowed for a function freg. We drop this for now, but will return to it shortly; as we will see, freg is not unique. To determine the residues Rn(s), we use the same technique as at tree-level. We have t=2Re+s2n [G1 loop(u; v)] = u +n An(v) log2(u) + Bn(v) log u + Cn(v) ; (3.24) where An; Bn; Cn are easily determined by plugging M1 loop of (3.23) into the Mellin amplitude formula (3.1). To x the Rn(s) we insist upon equality of An with the log2(u) term in (3.22). Given the Mellin representation (3.8) of the conformal blocks in the u 1 expansion, this xes the Rn(s) completely for every n. For example, the leading residue R0(s) is determined by the following equation: Using the Mellin representation (3.17) of g2coll +`;`(v) determines R0(s) to be (3.25) (3.26) HJEP07(21)36 1 `=0 R0(s) = X a0;` ( 0(1;`))2d ;` Q`;0( s; 2 ) ; (0) where d ;` was de ned in (3.18), and Q`;0( s; 2 ) is the polynomial (3.9) at intermediate twist 2 . Note that in the formula for R0(s), the coe cients of Q`;0( s; 2 ) are manifestly positive. Higher Rn(s) can, with some work, be extracted similarly. By matching Bn in (3.24) to the log u terms in (2.18), one can compute n(2;`), as we will show in an explicit example shortly. UV divergences and freg We now return to the physics of the function freg in (3.23). The rst point to note is that (3.23) is a solution to crossing for any permutationsymmetric freg. The minimal solution is freg = 0. Indeed, freg re ects the freedom to add a homogeneous solution to the second-order crossing equations (2.18). Such solutions sit in one-to-one correspondence with quartic contact interactions in AdS; in Mellin space, these are simply crossing-symmetric polynomial amplitudes [15, 51]. So we should think of freg as a choice of one-loop renormalization conditions for the quartic part of the e ective action for the light elds in AdS, dual to a choice of one of the in nite solutions to the one-loop crossing equations that di er by polynomials, i.e. nite local counterterms in AdS. What happens when the bulk theory is one-loop divergent? In this case, one must include in the bulk some diverging local counterterms to restore niteness. Due to their locality, these again appear in the function freg. This was explained in general terms in the Introduction; for more discussion, see appendix C. In the explicit results for scalar theories that will follow in section 5, we will see very nicely in detail how bulk UV divergences show up in the one-loop CFT correlators. For all of these reasons, freg is not unique, and may sometimes not be nite before renormalizing the bulk theory. We note that various high-energy limits, such as the Regge limit of large s and xed t < 0, may place some constraints on freg, see e.g. [48]. 3.3.2 Let us treat some simple and instructive examples. The rst is 4 theory in AdS. There is a single non-trivial one-loop diagram, the bubble diagram, in each channel. (There are also diagrams which lead to mass and wave function renormalization of , but these only serve to renormalize Mtree, which is anyway constant in this case.) On the CFT side, as explained earlier in section 2, there are only This matches (5.13). This is a substantial check on the match between CFT and AdS: we have successfully reconstructed the 4 one-loop amplitude from the conformal bootstrap. The expected UV divergence structure is apparent in the above: at large m, one has ^ Rm This leads to a divergence in the sum over m for d 3 | that is, in AdSD 4 | with a logarithmic divergence at the critical dimension dc = 3. We note for later that the d = 2 amplitude can be resummed to yield the t-channel amplitude n2`+1 ; n2`+2 ; for d = 2, for d = 4. It is not obvious that the low-spin 0(2;`) as computed from the Mellin amplitudes above will match those from the crossing problem. For this reason, we would like to analytically compute 0(2;`) for ` = 2; 4 directly from (5.14). This has never been done. Doing so requires new techniques that should be useful more generally for extracting anomalous dimensions from Mellin amplitudes with an in nite series of poles in a given channel. We devote section 6 to this endeavor. The end result is a perfect match for ` = 2; 4 in both d = 2 and d = 4. 5.1.4 Relation to lightcone bootstrap Note that in both d = 2 and d = 4, the anomalous dimensions are negative and monotonically increasing with `: 0(2;`) < 0 ; We have checked this behavior to higher ` as well. These properties must in fact hold for all ` and all (unitary) , as can be explained by resorting to Nachtmann's theorem and the lightcone bootstrap. The basic point is that, because 0;`>0 = 0, these one-loop anomalous dimensions are actually the leading corrections to the mean eld theory result. (1) The O O OPE is re ection positive and contains only even spin operators, which implies monotonicity via the arguments in [35, 36, 43]; moreover, the negativity follows from the large spin asymptotics given in (3.40). 5.1.5 More general contact interactions We could also consider more general solutions, where n(1;s) is di erent from zero also for s 6= 0. This corresponds to (@ )4-type theories, etc. Let us analyse the issue of divergences in this case. For instance, for s = 2 all the explicit results we have obtained (too cumbersome to be included here) are consistent with (5.20) On the other hand, n(1;s) has generally an enhanced behaviour, with respect to the solution studied above. For instance, an irrelevant interaction such as (@ )4 leads to a behaviour n;s (1) nd+1, see [60], as expected from the analysis of appendix C. For d = 2 this implies that the resulting 0(2;`) will be convergent only for ` > 2. For d = 4, the result will be convergent only for ` > 4. The four-point triangle diagram of AdS We now consider the following (Euclidean) AdSd+1 e ective theory: Lbulk = 1 2 m2 2 + 1 2 3! On the crossing side, we now consider in more detail the solution discussed in section 2, together with a truncated 4 solution with support only on operators with ` = 0. We will compute the crossed term contribution, proportional to This only receives spin-0 contributions, and computes the sum over channels of the fourpoint triangle diagram in AdS5, as shown in gure 3.14 For de niteness, we rst take d = 4 and = 2, hence m2 = ( d) = units. The rst-order data needed on the right-hand side is = 2 2(2 7(1 + n)2) 3 (1 + n)(3 + 4n(2 + n)) ; n;0 ; where n(1;0); 3 is computed in appendix D. We note that n(1;0); 3 (1); 4 n;0 leads to a convergent contribution for (5.26) even for the case ` = 0, as expected from the 1 for large n. This bulk. For the rst few spins we obtain In addition, the full theory will have a term proportional to 24 from the bubble diagram of gure 2, which is exactly as before, plus a term proportional to 43 from the box diagram of gure 1, which is harder to compute. 14For general eld theories on AdS, in which each vertex comes with an independent coupling constant, it is easy to identify the number of vertices of each type in the diagrams that contribute to each term in the crossing equation and in the anomalous dimensions. In speci c theories like supergravity, many di erent coupling constants are related and it is di cult to separate the contributions from individual diagrams; the solution to crossing just gives the full 1-loop amplitude. amplitude for a m2 = AdS3 is given in (5.41). hence m2 = 0 in AdS units. Now we use 4 scalar in AdS5 is given in (5.38). The amplitude for a massless scalar in The same analysis of (5.26) can be done also for d = 2. We again take = 2, and n;0 = 2(4n + 5) 2 3 (n + 1)(n + 2)(2n + 3) ; n;0 3 = 4 (2n + 3) : Analogously to the previous case, we compute the value of (2) for some value of the spin 4 case, we may use the 1=J expansion of 0(2;`) to reconstruct the bulk amplitude. The 1=J expansion used to derive the above results for d = 4 is 1 + 18 1 Analogously to the 4 case, we nd the following result: with M1 loop(s; t) = 1 X m=0 t Rm (4 + 2m) + (crossed) ; Rm = 3(10 + 7m)p (1 + m) 23 4 : ( 52 + m) (5.31) (5.32) (5.33) (5.34) (5.36) (5.37) The amplitude can be resummed: in the t-channel, say, as quoted in the introduction. This gives a prediction for the triangle Witten diagram for a m2 = 4 scalar in AdS5. Note the striking similarity to the 4 bubble diagram for d = = 2 in (5.21), which is completely unobvious from the spacetime perspective. We can perform the same analysis for d = 2 and = 2. The only di erence with respect to the previous case is n(1;0); 3 , which is computed in appendix D. The 1=J expansion is 1 + 2 1 25 J 2 8 1 In this case, we nd (5.36) with We can again resum the amplitude and obtain, in the t-channel, 6H(1 t 2 t 2 t where H(x) denotes the harmonic number of argument x, de ned for x 2= Z via the relation to the digamma function, H(x) = (x + 1) + . This gives a prediction for the triangle Witten diagram for a massless scalar in AdS3. 6 Computing anomalous dimensions from Mellin amplitudes In this section, we develop techniques for analytically computing double-trace anomalous dimensions from Mellin amplitudes. In particular, we focus on cases where the amplitude has an in nite series of poles. This necessarily occurs at one-loop as explained in this work, but also occurs in the tree-level exchange diagram of 3 for generic , or the tree-level exchange of a dimension 0 scalar between external dimension scalars, where 0 2 2Z. To our knowledge, the only treatments that have appeared in previous literature = 2 deal with nite sums of poles. As an application, we derive the one-loop 4 anomalous dimensions for ` = 2; 4, described in the previous section. 6.1 General problem the form Consider an exchange amplitude between identical external scalars of dimension , of for some residues Rm and some internal dimension 0. If this represents a tree-level 0 scalar primary, say, the residues are Rm(t) / Qm;0(t; ). I`( ; ) 2 s 2 2 2 2 s 3F2 The double-trace anomalous dimension 0;`>0 receives contributions from the two crossed channels: from (3.19), Rm(2 )I`( ; 0 + 2m) : (Recall that the direct-channel amplitude only contributes to ` = 0, since it evaluates to a constant on the pole at 2 .) We split the analysis into two parts. First, we evaluate I`( ; ), i.e. we determine the contribution to the anomalous dimension from a single pole. Next, we perform (6.3), summing over contributions from all poles. To evaluate I`( ; ), we close the contour to the left, picking up an in nite series of poles at s = 0; 2; 4; : : :. The resulting in nite sums can be regularized using Hurwitz zeta functions. Upon looking at several examples, one infers the following structure for (6.2) (6.3) 2 Z: (6.4) (6.5) (6.6) (6.7) (6.8) 2 X n=nmin I`( ; ) = Pn( ) 2 nmin = (` + 2 4) : n; 2 = Bn+1( 2 ) : n + 1 where The Pn( ) are degree-(n nmin) polynomials in , and P1( ) = 0. All but the 2; 2 terms reduce to Bernoulli polynomials in , Since Bn+1 is degree-(n + 1), we can rewrite the form of I` as I`( ; ) = P`+2 3( ) + R`+2 2( ) 2; 2 ; where Pm and Rm are polynomials of degree m.15 Note that (2; x) = 0(x) = d2x log (x). Now we want to sum over all poles at = 0 + 2m. Plugging (6.7) into (6.3), (1) 15There is some potential ambiguity in these polynomials; this can be xed in a given case by comparing to numerical integration. two regularization methods. P`+2 0 as xed. The second term is somewhat tricky. To proceed we employ The rst is an exponential regularization. This is useful when evaluating the sum over 1 X Rm(2 )P`+2 3(m)e m : 0( 0 + m) = 0 dt t e t( 0+m) 1 e t : Performing the sum and expanding near = 0, the prescription is to keep the nite term, dropping terms that are power law divergent. The second is an integral regularization. This is useful when evaluating the sum over 2(m) 0( 0 + m). Speci cally, we turn to the integral representation of 0( 0 + m), Swapping the order of the sum over m and the integral, performing the sum over m, and then performing the integration analytically, the prescription is to keep the nite term, dropping terms that are power law divergent near t = 0. We have checked that these two methods agree in several examples in which both can be carried to the end, e.g. a tree-level scalar exchange with = 2; 0 = 3. We now apply the above to compute 0(2;`) for ` = 2; 4. To make contact with the previous section, we take d = 4; = 2, and use the residues (5.11). From (6.3), (2) RmI`(2; 4 + 2m) : (6.11) First we compute I`(2; 4 + 2m). Let's rst focus on ` = 2. Closing the contour in (6.2) to the left, we pick up the poles at s = 0; 2; 4; : : :, which yields the following in nite sum: I2(2; 4 + 2m) = k=0 X1 (k + 1) 15k3 + 5k2(4m + 13) + k(40m + 86) + 22m + 38 6(k + m + 2)2 We can regularize this using Hurwitz zeta functions of the form (y; m + 2) for y = 2; 1; 0; 2. Comparing this result to numerical integration, we nd that an extra additive polynomial is required. The end result is where I2(2; 4 + 2m) ! P(m) + R(m) 0(m + 2) ; P(m) R(m) 1 36 1 6 (30m3 + 75m2 + 71m + 23) ; (m + 1)2(5m2 + 10m + 6) : (6.9) (6.10) : (6.12) (6.13) (6.14) We now need to perform the sum (6.11). We rst do the sum over P(m) using an exponential regulator. The sum yields a linear combination of generalized hypergeometric functions; upon expanding in small and keeping the nite term, we get 1 X m=0 P(m)Rme m 76 ! 105 2 : Next, we use the integral regularization on the sum over the R(m) term. After performing the sum inside the integral, we have 1 X m=0 R(m)Rm 0(m+2) = 0 t 1 e t Performing the integral, and keeping the nite terms, 1 X m=0 R(m)Rm 0(m + 2) ! 583 + 174 2 3465 2 : Adding this to (6.15) and multiplying by (-2) to obtain (6.11), the nal result is This agrees with equation (5.7). An analogous procedure can be carried out for ` = 4. The analog of (6.13) is with I4(2; 4 + 2m) ! P(m) + R(m) (2; m + 2) ; P(m) = R(m) = 1 60 630m5 + 2835m4 + 6195m3 + 7350m2 + 4579m + 1159 1800 Carrying out the sum over m using the above techniques, we get 1 X m=0 P(m)Rm ! 214500 104267 2 1 X m=0 R(m)Rm 0(m + 2) ! 2 83636 + 18825 2 1126125 Adding the two numbers and multiplying by (-2), this agrees with (5.8). We have repeated all of the above for d = 2, nding agreement there as well. To summarize, the results of this subsection give further con rmation that our solution to the crossing problem is equivalent to a direct computation of 4 one-loop Witten diagrams in AdS. We reiterate that this agreement acts as a check on a match between two independent techniques used to derive anomalous dimensions: on the one hand, the large spin resummation technique used in the crossing problem, without reference to any amplitude; and on the other, the techniques of this section used to extract low-spin data from M1 loop. (6.15) (6.16) (6.17) (6.18) (6.19) (6.20) 2 : (6.21) A remark on 3 theory There is a small subtlety when computing leading pole of an exchange diagram in a pole at twist in all three channels is n(1;`) for 0 = . A common example is for the 3 theory. The fully symmetrized contribution of s 1 If we evaluate the s and u^ poles on the 2 double-trace pole at t = 2 , they cancel. Thus, naively, so do the s- and u^-channel contributions to 6 add, just as they do for . To get around this, one can simply deform the internal dimension by a small amount, + , perform the computation in which the two example, see appendix D for the computation of n(1;`) in 3 theory for = 2. channels add, and then take ! 0. A spacetime computation con rms this result. For (1) n;` . However, they are supposed to In this paper we initiated an analysis of large N CFT four-point correlators at next-toleading order in 1=N , which map by the AdS/CFT correspondence to one-loop diagrams in AdS space. We presented general methods to analyze correlation functions at this order, and implemented them explicitly for two examples: a 4 theory in the bulk, and a triangle diagram in a 3 + 4 theory in the bulk. There are various levels of extension of what we have done here, most of which are needed in order to study the 1=N expansion in more generic, full- edged holographic CFTs. We rst discuss some of these, and then move on to broader future directions. 7.1 An immediate priority, and a necessary step toward solving bona de CFTs, is to solve the crossing equations when the OPEs contain single-trace operators. In the single-scalar theory, this would yield a computation of the scalar box diagram in AdS 3 theory. We could also allow exchanges of operators with spin. These should present some technical complications, but we do not expect them to lead to any qualitative changes. A particularly important version of this is to incorporate the stress tensor, which allows us to access graviton loops.16 One would also like to extend our methods to include multiple species of operators, as in [71]. Indeed, a generic CFT, as opposed to a bottom-up generalized free eld theory, always has an in nite number of single-trace operators. If there are additional elds in the bulk with four-point couplings ~4 2 2, then the corresponding bubble diagrams are also easy to compute, given the tree-level hOOO O i; see [26]. When 16Note that for operators of 2 Z, there is potential mixing between [OO] and [T T ], at least in some channel where global symmetry-singlet [OO] operators contribute. there are also three-point vertices, the situation is more complicated. The case when some [O O ] operator is degenerate with a [OO] operator with the same quantum numbers is discussed in appendix A.1, and requires generalizing the bootstrap analysis to di erent external operators. It should be straightforward to understand the form of M1 loop when external dimensions are unequal. The basic structure will be identical to (3.23): M1 loop will have poles at = 2 + 2n in every channel, with residues xed by rst-order data. An extension to higher loops would also be pro table. On the CFT side, one will generally have to contend with triple- and higher-trace operators. By the arguments of section 3, one sees that at O(1=N 2(L+1)) | dual to L-loop order in AdS | ML loop has poles of degree information from L. In general theories, going to O(1=N 6) requires additional ve-point functions. However, in theories that have a O $ O symmetry, these ve-point functions vanish (and correspondingly no triple-trace operators appear in the O O OPE). So in these theories it may be possible to compute the four-point functions and anomalous dimensions also at two-loop order, with no new conceptual wrinkles. What we really seek, however, are the AdS loop-level Feynman rules for Mellin amplitudes, thus giving an algorithm for any L-loop calculation. In general the 1=N expansion is only asymptotic, and there are non-perturbative e ects scaling as (say) e N that must be understood before even attempting to continue the large N results to nite values of N . Can we use our methods also for such non-perturbative contributions? As we discussed, the crossing analysis simpli es considerably when the dimension is an integer. It would be interesting to analyze the case of non-integer and to obtain explicit results for this case as well. For theories with three-point vertices, we found (see (2.30)) that the four-point function at O(1=N 4) has a contribution proportional to u log2(u) log2(v), with a crossing-symmetric coe cient function h(u; v). What are the form and content of this function? Our analysis in this paper did not assume any additional symmetries. It should be simple to take into account additional global symmetries. Incorporating supersymmetry should also be straightforward, at least in principle, with superconformal blocks replacing the conformal blocks. An especially interesting example, as always, is the d = 4, N = 4 SYM theory. In the 2 Y M = gY M N ! 1 limit, the bulk theory only contains the elds dual to protected single-trace operators. The solution to crossing in this limit at order 1=N 2 was performed in [39]. Its generalization to order 1=N 4 involves all the issues mentioned earlier: in particular, there is an in nite number of single-trace operators, and they all have integer dimensions so that there can be complicated mixings between the HJEP07(21)36 various double-trace operators. Luckily, the four-point functions of all these protected single-trace operators were recently computed in [66], and this information should be su cient to work out the mixing matrix, and thus to compute the correlation functions of protected operators in this theory at order 1=N 4. It would be interesting to perform this analysis. In fact, the analysis should be simpler than it may appear. We now make a potentially powerful observation. Consider the Mellin amplitude for the four-point function hO200 O200 O200 O200 i. At large Y M , all single-trace operators in the O200 O200 OPE have even twist: these are the operators Ok with even k BPS operators in the [0; k; 0] representation of SU(4). (O200 2, which are the 1/2 O2.) Due to nonrenormalization of the Ok dimensions and SU(4) selection rules, the only operators appearing in M1 loop will be the double-trace operators [OkOk]n;`, with k 2; so all poles in M1 loop sit at even twist = 4 + 2n. Now, for every value of n, there are double-trace operators [O200 O200 ]n;`. If we compute M1O2l0o0oOp200 for this correlator | that is, the piece of M1 loop xed by requiring a match to the contributions of the [O200 O200 ]n;` operators, as done in section 3 | the residues at all twists = 4 + 2n are xed by [O200 O200 ]n;` tree-level data. Therefore, up to regular terms, this xes the full one-loop amplitude! That is, up to regular terms, M1 loop = M1[O2lo00oOp200 ] : (7.1) While the operator mixings mentioned above still plague the calculation, (7.1) says that the full one-loop amplitude | which involves an in nite set of diagrams involving virtual Ok loops | is determined just by the anomalous dimensions [O200 O200 ]n;` operators. This is a great simpli cation, apparently due to the e ect of n(1;`) for the maximal supersymmetry on the spectrum. In general, for a four-point function of some operator O, this coincidence of poles occurs whenever O 2 Z and the spectrum of twists in the O Besides N = 4 SYM, this also occurs when O is the bottom component of the stress tensor multiplet of the d = 6, N = (2; 0) theory of M5-branes. We can also analyze many other interesting supersymmetric conformal eld theories, such as the d = 3, N = 8 theory of M2-branes (which does not have the same simpli cation described above). In both of these cases, there is again a gap to the non-protected operators, but here it scales as a power of N that does not involve an extra independent parameter. Thus one cannot separate the loop expansion and the derivative expansion in the bulk.18 In any case, the loop diagrams in the dual AdS bulk can still be computed O OPE is even.17 by the methods described in this paper. 17This phenomenon has a tree-level version: in a tree-level exchange of twist between external operators of dimension , the amplitude has only a nite number of poles when 2 2 2Z. This happens because the single-trace and double-trace poles collide, and would thus produce a triple pole, violating the 1=N expansion, unless these single-trace poles drop out of the amplitude. This was also recently noted in [66]. 18At some speci c low orders in 1=N , it is possible to separate the di erent contributions to the correlation functions: in particular, the leading 1=N correction is due not to a loop, but to a higher-derivative correction to the action that descends from anomalies in d = 11 supergravity (e.g. [72]). 7.2 When we have a standard eld theory in AdS space (as opposed to a gravitational one), it has not just correlation functions with sources at the boundary as we discussed in this paper, but also correlation functions of operators at arbitrary bulk points. Are these determined in terms of the correlation functions with boundary sources? Can we say anything about them by our methods? In our discussion of the N = 4 SYM theory we integrated out the stringy states, but we can repeat the same story when including non-protected string states. For nite Y M there are additional operators contributing with dimensions at least of order 1=4 . These operators can be integrated out in an expansion in 1= 1Y=M4 , whose form Y M at order 1=N 2 was discussed in [39]. Using this information it should be possible to work out also the order 1=N 4 correlators in a systematic expansion in 1= 1Y=M4 . Can we use the large N expansion of the crossing equation to learn anything about the non-protected states? For bulk theories which are string theory backgrounds, the 1=N expansion (the loop expansion in the bulk) coincides with the genus expansion of the worldsheet theory. The correlators we discuss arise as integrated correlation functions in this worldsheet theory. What does our analysis teach us about these worldsheet theories? Can we relate the crossing equations in the CFT and in the worldsheet theory? We close with some words on the relation of the large N bootstrap to at space physics. An alternative way to approach the AdS amplitudes problem might have been to start from known facts about S-matrices, and nd analogs or extensions to AdS. We took a di erent tack, but it would be very interesting to turn to these questions using our results. In [26], the emergence of the optical theorem in the at space limit was studied, but one would also like to know whether there is a direct analog at nite AdS curvature. Similarly, in our one-loop crossing computations we found speci c harmonic polylogarithms appearing. As we noted, this suggests an intriguing underlying structure akin to at space amplitudes. On the other hand, the one-loop Mellin amplitudes themselves were given by the more familiar generalized hypergeometric functions and, in the case of (5.41), a digamma function. What class of functions forms a basis for the multi-loop solution of the crossing equations, and for the AdS Mellin amplitudes themselves? Which diagrams form a basis for all others at a given loop order? The answers would presumably be closely related to the possible existence of AdS analogs of generalized unitarity, on-shell methods and the like. It would be fascinating to try to understand the big picture here. Finally, we note that Mellin amplitudes admit at space limits [15]. If one can develop the solution to crossing to successively higher orders in 1=N , taking that limit would shed light on at space higher-loop amplitudes. A speci c, and di cult, longer-term challenge in the supergravity community is to determine the critical dimension above HJEP07(21)36 which the four-point, ve-loop amplitude in maximal supergravity diverges. This has resisted years of direct attack using advanced methods [73{75]. It would be fascinating if, eventually, the ve-loop crossing equations, applied to the holographic dual of gauged maximal supergravity, could be employed in this endeavor. Acknowledgments We wish to thank Nima Afkhami-Jeddi, Michael B. Green, Vasco Goncalves, Tom Hartman, Zohar Komargodski, David Simmons-Du n, Ellis Ye Yuan and Sasha Zhiboedov for helpful discussions. The work of OA was supported in part by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12), by an Israel Science Foundation center for excellence grant, by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by a Henri Gutwirth award from the Henri Gutwirth Fund for the Promotion of Research, and by the ISF within the ISF-UGC joint research program framework (grant no. 1200/14). OA is the Samuel Sebba Professorial Chair of Pure and Applied Physics. The work of LFA was supported by ERC STG grant 306260. LFA is a Wolfson Royal Society Research Merit Award holder. AB acknowledges the University of Oxford for hospitality where part of this work has been done. AB is partially supported by Templeton Award 52476 of A. Strominger and by Simons Investigator Award from the Simons Foundation of X. Yin. EP is supported by the Department of Energy under Grant No. DE-FG02-91ER40671. A Operator content of the one-loop crossing equations In this appendix we discuss the operators that can appear in the OPE of two identical single-trace primary operators O and O of dimension expansion they contribute to the crossing equation. The upshot is that at order 1=N 4, we do not have to consider any operators with more than two traces appearing in the OPE. , and at which order in a large N The notation is that [O1O2 : : : Om] is an m-trace primary operator corresponding to an m-particle state in the bulk (and appearing at N Om(xm); for the precise de nition at m = 2, see appendices of [15, 27]). All operators will be normalized such that their two-point function is one. We will choose a basis in which there is no mixing between operators with a di erent number of traces (we will discuss mixings of di erent double-trace operators below). This means, for instance, that [OiOj ] is not exactly the operator appearing in the OPE of Oi and Oj , but may di er from it at order 1=N ; these di erences will not be important in the order we work in. On general grounds, connected n-point functions of single-trace operators scale as 1=N n 2 in the large N limit. Naively this implies that the OPE coe cient of a k-trace operator, proportional to hOO[O1 Ok]i scales as 1=N k. In general this expectation can fail only if there is an extra disconnected contribution to this correlation function. However in our case, since we chose the single-trace operators to be orthogonal to operators with more traces, such a disconnected correlation function can only appear for the operators [OO], which have OPE coe cients of order one. Thus the OPE coe cient of operators 1 in the OPE of with three or more traces is suppressed at least by 1=N 3, so they will not contribute to the crossing equation at order 1=N 4. The only operators contributing at order 1=N 4 are then: Single-trace operators O1, with some even spin ` (a special case is the energymomentum tensor): the OPE coe cient cOOO1 is generically of order 1=N , so they contribute to crossing already at order 1=N 2. At order 1=N 4 we will see corrections to these contributions due to 1=N 2 corrections to the dimensions of O and O1, and to cOOO1 . These cannot be determined by crossing since they are the basic inputs | in the bulk these are masses and three-point vertices that need to be determined by some renormalization condition at all orders in 1=N . Thus from the point of view of the crossing equation we need to take these as given. If we use renormalization conditions that are independent of N , and in particular for protected operators in superconformal eld theories, single-trace operators will appear in the crossing equation only at order 1=N 2; otherwise their contributions at higher orders are simply related to the leading order contribution and to the corrections to the dimensions and single-trace OPE coe cients. coe cients a(n0;`), and with dimensions 2 Double-trace operators [OO]n;`: these appear already at order 1 with squared OPE + 2n + `. As we discuss extensively, at higher orders in 1=N they give contributions related to the corrections to the OPE coe cients and dimensions of these double-trace operators. Other double-trace operators [O1O2]n;l: the OPE coe cient cOO[O1O2]n;l is of order 1=N 2. Thus, generally these operators appear in the crossing equation at order 1=N 4, with a contribution depending on the leading order dimension 1 + 2, and on the leading order cOO[O1O2]n;l . The latter depends on four-point couplings in the bulk which are arbitrary, so from the point of view of the four-point function hOOOOi they will give us parameters that we cannot determine. However, because these contributions depend only on the leading order dimensions, they generically do not come with any logs in the direct channel, so they will not a ect the universal terms that we discuss in this paper; they give rise to independent poles in Mellin space. This is not true when these operators mix with the [OO] operators, as we discuss below. At order 1=N 6 the analysis will change, and in particular triple-trace operators will also start appearing, depending on (undetermined from crossing) ve-point vertices in the bulk. A.1 Degeneracies One important issue that was ignored in the analysis above is mixing between di erent double-trace operators when they are degenerate; this often happens in interesting examples, and a mixing of [OO] with other double-trace operators signi cantly modi es the analysis. As a typical example, consider a 2 21 eld theory on AdS, where and 1 are scalars with the same mass, and where O is dual to and O1 to 1. In this theory, the OPE of HJEP07(21)36 The resolution is that the two double-trace operators mix: there is a bulk tree-level O and O contains [OO] starting at order 1 from the disconnected diagram in AdS, and [O1O1] starting at order 1=N 2 from an X-shaped diagram, and no single-trace operators. There are no tree-level diagrams contributing to hOOOOi and to hO1O1O1O1i, so naively the analysis at order 1=N 2 implies that [OO] and [O1O1] have no anomalous dimensions at this order. We then expect to have no logarithmic terms in the direct-channel four-point function at order 1=N 2, and no double-logs at order 1=N 4 (i.e. no poles in M1 loop). But on the other hand, the one-loop diagram contributing to hOOOOi is clearly the same as in the 4 theory, which does have such double-logs/poles since the latter theory does have a non-trivial tree-level diagram. operator (1) = B diagram giving a non-zero h[OO][O1O1]i diagonal two-point functions is OA turns out that the rst operator has an anomalous dimension 1=N 2. The correct basis of operators with [OO] + [O1O1] and OB(1) = C=N 2 and the second [OO] [O1O1], and it A C=N 2, for some constant C. This reproduces the two-point functions at order 1=N 2. It also explains why we get double-logs at order 1=N 4 (poles in M1 loop) in hOOOOi, since these are proportional to ( A(1))2 + ( B(1))2 (both OA and OB appear in the O O OPE), which is non-zero. The lesson is that in general, we have to be careful of double-trace mixings; all operators that mix with [OO]n;` appear in the crossing equation already at order 1=N 2, and will lead to double-logs at order 1=N 4. The coe cients of these double-logs cannot be computed without knowing the precise mixing matrix: one has to know all correlators h[OO][O1O2]i at order 1=N 2, which can be extracted from tree-level hOOO1O2i four-point functions, before one can use the crossing equation at order 1=N 4. Note that mixings of this type occur in the N = 4 SYM theory, complicating its analysis. B Explicit expansions In this appendix we display explicit results for the large spin expansion for 0(2;`) in several examples. Recall that the total result is the sum over contributions from each conformal block in the dual channel. In a case in which the solution at order 1=N 2 has support only for spin zero we obtain (2) 0;` (n;s) (0) cn;s (2) J 2 +2n ^0;` (n;s) : (` + 2)2(`(`(6`(` + 7) + 115) + 148) + 77) + 3J 6 3`2 + 9` + 8 (2)(` + 1) ; (` + 1)(` + 2)3(`(`(3`(`(10`(` + 10) + 443) + 1111) + 4975) + 4187) + 1558) + J 5 8 5`4 + 30`3 + 79`2 + 102` + 54 (2)(` + 1) ; ^0;` (n;0) For ^0;` (0;0) ^0;` (1;0) ^0;` (2;0) 3 2 5 36 6 0(2;`) = 8 1 X a(n0;0) (1) 2 (2) 0;` (n;0) has the structure explained in section 4. = 2 in d = 4 we obtain for the rst few cases 2(` + 2)(`(` + 4) + 5) ` + 1 + 2J 4 (2)(` + 1) ; (B.1) (B.2) where we have introduced J 2 = (` + 1)(` + 2). As explained in the body of the paper, with these ingredients it is possible to obtain zero, we get 0(2;`) also for nite values of the spin `. For spin 36(1 + n)4(5 + 2n(3 + n)) (3 + 4n(2 + n)) 72(1 + n)8 3 + 4n(2 + n) (2)(n + 1) 2 : (B.3) = 2 in d = 2, equivalently one can nd the form of ^0(2;`)j(n;0) and compute 0(2;`). As already discussed, this sum is divergent. For spin zero in this case we obtain 0(2;0) = X 9(19+n(25+2n(6+n))) 2(1+n) better organised in powers of J 2. For instance, for the interaction 4 with For each model we can obtain the expansion of 0(2;`) around large `. The expansion is = 2 in d = 4 + 9(1+n)2(2+n)2 2 = 2 : (B.4) HJEP07(21)36 18 1 5 J 2 + 96 1 7 J 4 + 360 1 7 J 6 + 74304 1 724320 1 1001 J 10 + 2 ; (B.5) for the interaction 4 with = 2 in d = 2 we obtain 1 + 4 1 5 J 2 + 4 1 7 J 4 + 16 1 35 J 6 + 16 1 55 J 8 + 1856 1 5005 J 10 + while for the mixed interaction 33! 3 + 44! 4 with = 2 in d = 4 we obtain 96 1 72 1 576 1 The expansions above are asymptotic. In the body of the paper we have shown how to resum the expansions and compute them for nite values of the spin. It is interesting to compare the asymptotic series above with the correct results for di erent values of the spin. For instance, for ` = 2 we have J 2 = 12. Including the rst six terms shown above for 4 and 3 + 4 in d = 4 we would obtain 2 ; (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (2) 0;2 (2) 0;2 0:11977 2; 0:591144 32 4; for 4 for 3 + 4 to be compared with the exact values (174 2 ; (39 2 350) 32 4 0:591146 32 4; for 3 + 4 : We see that the values we obtain from the asymptotic series are remarkably close to the correct values, even for spin two! In the case of convergent answers, even the approximation for spin zero is very good. When we compute bulk loop diagrams we expect to get UV divergences. Since these arise at short distances, they should take a similar form in AdS as in at space, and at any loop order we should be able to cancel them by local counter-terms in AdS. In general the bulk theories we discuss are e ective theories which are non-renormalizable, so they require a cuto , and at higher orders in perturbation theory we will need to add more and more counter-terms, but in this paper we just discuss the one-loop order. As argued in the Introduction, in our bootstrap computation related to a divergent bulk diagram we expect to nd a divergence in n(2;`), and we expect that when we regularize it (for instance by putting some cuto on the sums), the divergence is precisely proportional to from some local bulk terms, so that it can be removed by putting in appropriate cuto n(1;`) coming dependent bulk terms. Recall that on general grounds we expect any local bulk term that is allowed by the symmetries to appear with an arbitrary coe cient, both from the bulk point of view, and from the bootstrap point of view, since any such term gives a solution to the crossing equations. Thus at any loop order any solution that we nd for the four-point function, both from the eld theory and bootstrap points of view, is just up to bulk terms. This means that we should take both the dimensions and the three-point functions of single-trace operators, at all orders in 1=N , to be inputs to the computation, that we cannot determine just from the crossing equations in a 1=N expansion. In addition we have a freedom to choose any local four-point terms, namely to shift the solution by any of the \homogeneous" solutions to the crossing equations that correspond to nite-order polynomials in Mellin space (we called them freg in section 3.3.1). We expect to need this freedom in order to cancel divergences. We cannot x it just from crossing. Consider rst the 4 theory in AdS5. The coupling constant here has dimensions of length, and one can de ne a dimensionless coupling =RAdS, that in our 1=N expansion is proportional to 1=N 2. In at space the four-particle tree-level scattering amplitude goes like ; when we translate it into some dimensionless quantity this will go at high energies as E where E is a typical energy. In AdS the role of the energy is played by n, so we expect to nd for the tree-level four-point amplitude a result going as (1) n;` / n ' N 2 (C.1) at large n, which is indeed what we nd (5.1). (In this case the answer happens to vanish for l > 0.) Note that large n here means n 1 and n , so that the energy is larger than the mass and the scale of the AdS radius. At one-loop in at space we have a linear divergence, and the amplitude with a nite cuto goes at high energies as 2 + E + ). Note that we do not get a logarithmic divergence; indeed such a divergence would multiply E but there is no local counter-term that could cancel this (higher-derivative couplings in the bulk give higher powers of E). Noting that the divergence is just a constant, it can be canceled by shifting by a term proportional to 2 . Translating to AdS as above, we expect to nd for the one-loop, four-point function at large n n;` / 2(n2 + ~ n) ' n2 + ~ n N 4 where ~ is some cuto large n behavior of n(2;`) in 4 that we use to obtain a nite result. This is a prediction for the . We expect from the locality of the divergence that we could obtain a nite result by shifting n(2;`) by a term proportional to ~ n(1;`) of the 4 theory. Note in particular that this means that only ` = 0 terms should diverge, and this is indeed what we nd in section 5. Note also that as far as the crossing equations in the 1=N expansion are concerned, there is no obvious way to x the nite local 4 bulk term remaining after this subtraction. In general we get precise predictions for which divergences we should get in our computation. It should always be possible to cancel divergences in 4- counter-terms by adding terms proportional to the n;` 's that are associated with the counter-terms we need in the bulk. In Mellin space these divergences should always be a polynomial, of a nite degree related to the loop order. Above one-loop, divergences (1) related to counter-terms with more 's can also appear. The analysis of the N = 4 SYM theory, and the related supergravity on AdS5, is analogous. The only di erence is that we have to be careful if we regularize our computation, that the regularization preserves supersymmetry, otherwise we will get divergences that are related to bulk counter-terms that are di erent from the supersymmetric local terms in the bulk. Using a supersymmetric regularization the divergences should all be proportional to the tree-level contributions analyzed in [39]. We leave a detailed discussion of this case to n(k;`) that are related to Finally, if we consider the 3 theory in AdS5 (d = 4), the theory is super-renormalizable so there should be no divergences in the four-point functions that cannot be swallowed into the masses and three-point couplings in the bulk. In this case dimensional analysis implies that all n(k;`) should not grow at large n, and that we would not encounter any UV divergences in their computation. For d > 5 the bulk theory is non-renormalizable, so we expect the large n behavior of the tree-level terms to go as nd 5, and one-loop terms to go as the square of this. D 3 OPE data In this appendix we derive the tree-level anomalous dimensions of double trace operators due to a fully symmetric exchange of a scalar operator of = 2, i.e. for a = 2 scalar with a 33! 3 coupling in AdS. This result was quoted in (5.27) for ` = 0. Such an exchange has been considered in [11, 15] and it can be reduced to G(u; v) = (d)u2D1212(u; v) ; (D.1) D i (u; v) = 2 i ( i ) i i 2) x123 1 x224 2 where (d) is a constant which depends on the number of space time dimensions, in particular (4) = 8 23 = 2C2 and (2) = 2 23 = C2 OOO. The functions D(u; v) are de ned as 1 2 3 2 2 4 2 x13 x214x34 2 D i (xi) ; (D.2) 2 x14 x213x34 2 HJEP07(21)36 G(u; v) = G(u; v) + G = (d)u2 D1212(u; v) + D2211(u; v) + D1221(u; v) ; where in the last line the symmetry properties of D(u; v) have been used. To compute the anomalous dimension it is enough to focus on the terms proportional to log u in (D.8) and perform the conformal partial wave expansion G(u; v)jlog u = 2 1 X an;` n;` (0) (1); 3 un+2g4+2n+`;`(u; v) : Notice that the small u expansion of G(u; v) starts at order u, but this contribution does not contain any log u. This is consistent with the expectations, since the OPE contains the scalar operator of exact dimension two and all its descendants. It is straightforward to extract the anomalous dimension from (D.9), both for d = 2 and d = 4: = D 3 2 1 4 (v; u) = u 2 D 4 2 3 1 u 1 v v + u2G 1 v u u u 1 v v 1 v u u : (D.4) (D.6) (D.8) (D.10) HJEP07(21)36 where The symmetry properties of D(u; v) are D i (xi) = 2 i i i ( i 2) Z 1 Y dtiti i 1e 21 Pi;j titjxi2j : D 1 2 3 4 (u; v) = v 2 D 1 2 4 3 We would like to study the fully symmetrized amplitude which corresponds to d = 2 : d = 4 : ( 2(5+4n) (1+n)(2+n)(3+2n) 3 4 (`+1+n)(`+2+n) 3 2(2 7(1+n)2) (1+n)(3+4n(2+n)) 3 (`+1)(`+2+2n) 3 2 ; ` = 0 2 ; ` 6= 0 2 ; ` = 0 2 ; ` 6= 0 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] J.M. Maldacena, The large-N limit of superconformal eld theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. [2] S.S. Gubser, I.R. 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Ofer Aharony, Luis F. Alday, Agnese Bissi, Eric Perlmutter. Loops in AdS from conformal field theory, Journal of High Energy Physics, 2017, 36, DOI: 10.1007/JHEP07(2017)036